The objectives of the proposed research are two: (1) to contribute to the understanding of chance mechanisms in biological and health phenomena, and (2) to develop optimal statistical methodologies for analysis of experimental and observational data. The problems treated in the past and those contemplated for the future include the following: (1) Mechanisms of carcinogenesis induced by a variety of agents, radioactive or chemical, e.g. (new problem): Does the pronounced dose-rate effect of urethane lung tumorigenesis in mice depend solely on the limited speed of metabolism or, as claimed by Shimkin and Kauffman, must one assume the urethane and time dependent density of especially sensitive cells? While the available data refer to urethane, presumably similar mechanisms operate with other chemicals. (2) Problem of optimal multicomparison methodology, e.g. given a factorial experiment comparing cancer frequencies in animals treated with agents A, B, and C with that of controls, and given a small number alpha, say alpha equals 0.05, to determine methodology ensuring the probability of at least one error in the assertion that (for example) agents A, B. and C taken singly are "innocent," that their combination ABC is carcinogenic, and that other combinations, AB, AC, BC, are doubtful. (3) As recently found by Clifford, even the serial sacrifice experimental methodology (representing a most valuable innovation by A. C. Upton) is insufficient to provide information on all the transition probabilities from one combination of pathological states to another. The problem is to determine the assumptions (perhaps verifiably by some special experiments) under which all the relevant questions regarding synergisms (etc.) could be answered.